Question

Two circles have the following equations `(x -1 )^2+(y -4 )^2= 25` and `(x +3 )^2+(y +3 )^2= 49`. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

Answer 1

The 2 circle overlap but neither are contained within the other...

Explanation:

Given: Two circles
`C_1: (x-1)^2 +(x-4)^2=25`
`C_2: (x+3)^2 +(x+3)^2=49`

Required: Find if `C_2` contains `C_1` and the greatest distance between them? Check if the circles are contained with one another?

Solution Strategy:
A) Find the center of the circles
B) Use distance formula to find the separation distance of the centers
C) Compare to the sum of the radiuses, to distance of seperation
- If `r_(12) lt |r_(C_1)+r_(C_2)| and r_(12) lt |r_(C_2)|` Contain
- If `r_(12) lt |r_(C_1)+r_(C_2)|` circles overlap but no containment

`color(red)((A))` The centers of the two circles are:
`C_1: O_(c_1)(1,4; 5)` this reads center at (1,2) with radius r= 5
`C_2: O_(c_1)(-3,-3; 7)`

`color(blue)((B))`
Distance formula between two points `C_1(x_1,y_1) and C_2(x_2,y_2)` is given by:
`r_(12)=sqrt[(x_1-x_2)^2+(y_1-y_2)^2 ]`
`r_(12)=sqrt[(1-(-3))^2+(4-(-3))^2]=sqrt[(4)^2+(7)^2 ]~~8.06`

`color(green)((C))`
Is `r_(12) lt |r_(C_1)+r_(C_2)|`;
`8.06<(7+5)=12` True but `8.06> 7` so just overlap